Again this is super hard to prove if you assume God doesn't exist but I gave it my best shot:
1. To live is to know suffering (premise)
2. Suffering is not good (premise, definition)
3. If we know what suffering is, we must also know what it is not (argument from opposites) (if you don't know what this is it essentially states that things cannot be known without also knowing their opposite or privation i.e. hot cannot be known without cold)
4. The opposite of suffering is "not suffering," which is good. I will call it "happiness" (2,3)
5. We must know what happiness is (3,4)
6. In order for us to know what happiness is, it must exist (argument by definition. As I stated earlier, things which do not exist cannot be known, therefore things which exist must at least be knowable)
7. If there is an inherent suffering to life then there must be an inherent happiness to life as well (1,3,6)
8. The only reason we suffer is because we also have the capacity to be "happy" or not suffer (7)
9. Therefore, there is an inherent goodness to life because there is an inherent capacity to be happy, which is good. (8)
10. Non-living or "non-existence" has no inherent goodness nor badness becuse it is nothing, and "nothing" cannot have properties such as good or bad, thus there is no suffering nor happiness (definition of "nothing") (9, conclusion)
11. Living (existence) is better than non-living (non-existence) because living has an inherent goodness to it which non-existence does not. (10, Final conclusion)
Comment got too long, I am going to post a few notes in a second comment though